Candles on ridiculously sensitive scales

If we had the capacity to measure masses this small:
Would a candle in a closed jar be just as heavy as identical candle in a jar, but which is on fire?
(say, some chemical touched and ignited the candle in one jar, but is just sitting next to the candle in other, so mass is exactly the same)
As I understand, more energy in a system actually makes it heavier as particles move around and become relatively more massive, but that is if energy is put in to the system from outside. What happens in this case, where the energy is actually in the system from beginning, but in a form of chemical bounds?

As I understand, more energy in a system actually makes it heavier as particles move around and become relatively more massive, but that is if energy is put in to the system from outside. What happens in this case, where the energy is actually in the system from beginning, but in a form of chemical bounds?

since the energy is already there the system is neither gaining nor losing any energy. (assuming no light or heat escapes the burning jar)